Thomas Calculus 13th [Solutions]

796 Chapter 10 Infinite Sequences and Series
3. diverges by the nth-Term Test since
exist
n n
lim lim ( 1) tanh lim ( 1) 1
n
a e
n
n lim ( 1) does not
2n
n n b 1 e n
2n
n
ln n!
4. converges by the Direct Comparison Test: n! n ln n! n ln ( n) ln ( n)
n log n n!
n
log n n ! 1 , which is the nth-term of a convergent p-series
3 2
n n
5. converges by the Direct Comparison Test: a 12 1 2 12 2 3 1 2 12
1 a
2 2 3 4 a
2 3 4 5 3 4
2
(1)(3)(2) (2)(4)(3) (3)(5)(4)
a 4 3 4 2 3 1 2 12 , 1
12
5 6 4 5 3 4 2
(4)(6)(5) ( n 1)( n 3)( n
n 1
represents the given series and
2
2)
12 12 , which is the nth-term of a convergent p-series
2 4
( n 1)( n 3)( n 2) n
an
1
6. converges by the Ratio Test: lim lim n 0 1
n
an
n
( n 1)( n 1)
7. diverges by the nth-Term Test since if an
L as n , then
1 2 1 5
L L L 1 0 L 0
1 L
2
8. Split the given series into 1 and 2n
; the first subseries is a convergent geometric series and the
2n
1
2n 3
3
n 1
n 1
second converges by the Root Test: lim 2 lim n n
n n 2 n 11 1
2
3
9 9 9
1
n
n
n
3 3 (4)
9. f ( x) cos x with a f 0.5, f , f 0.5, f , f 0.5;
3 3 3 2 3 3 2 3
3
2
3
3
cos x 1 x 1 x x
2 2 3 4 3 12 3
10. f ( x) sin x with
(4) (5)
a 2 f (2 ) 0, f (2 ) 1, f (2 ) 0, f (2 ) 1, f (2 ) 0, f (2 ) 1,
(6) (7)
f (2 ) 0, f (2 ) 1; sin x ( x 2 )
3 5 7
( x 2 ) ( x 2 ) ( x 2 )
3! 5! 7!
11.
2 3
x
e 1 x x x with a 0
2! 3!
12. f ( x) ln x with
ln x ( x 1)
13. f ( x) cos x with
(4)
a 1 f (1) 0, f (1) 1, f (1) 1, f (1) 2, f (1) 6;
2 3 4
( x 1) ( x 1) ( x 1)
2 3 4
(5) (6)
f (22 ) 0, f (22 ) 1;
(4)
a 22 f (22 ) 1, f (22 ) 0, f (22 ) 1, f (22 ) 0, f (22 ) 1,
1 2 1 4 1
6
cos x 1 ( x 22 ) ( x 22 ) ( x 22 )
2 4! 6!
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